A statistical method for removing unbalanced trials with multiple covariates in meta-analysis

In meta-analysis literature, there are several checklists describing the procedures necessary to evaluate studies from a qualitative point of view, whereas preliminary quantitative and statistical investigations on the “combinability” of trials have been neglected. Covariate balance is an important prerequisite to conduct meta-analysis. We propose a method to identify unbalanced trials with respect to a set of covariates, in presence of covariate imbalance, namely when the randomized controlled trials generate a meta-sample that cannot satisfy the requisite of randomization/combinability in meta-analysis. The method is able to identify the unbalanced trials, through four stages aimed at achieving combinability. The studies responsible for the imbalance are identified, and then they can be eliminated. The proposed procedure is simple and relies on the combined Anderson-Darling test applied to the Empirical Cumulative Distribution Functions of both experimental and control meta-arms. To illustrate the method in practice, two datasets from well-known meta-analyses in the literature are used.


INTRODUCTION
Rationale 3 Describe the rationale for the review in the context of existing knowledge.Rows: 42-97 Objectives 4 Provide an explicit statement of the objective(s) or question(s) the review addresses.Rows: 98-99

Eligibility criteria 5
Specify the inclusion and exclusion criteria for the review and how studies were grouped for the syntheses.Rows: 164-186 180-185 Information sources 6 Specify all databases, registers, websites, organisations, reference lists and other sources searched or consulted to identify studies.Specify the date when each source was last searched or consulted.

Search strategy 7
Present the full search strategies for all databases, registers and websites, including any filters and limits used.
Selection process 8 Specify the methods used to decide whether a study met the inclusion criteria of the review, including how many reviewers screened each record and each report retrieved, whether they worked independently, and if applicable, details of automation tools used in the process.

Data collection process 9
Specify the methods used to collect data from reports, including how many reviewers collected data from each report, whether they worked independently, any processes for obtaining or confirming data from study investigators, and if applicable, details of automation tools used in the process.
Data displayed in the reports were organized in Excel sheets, and then set in two tables (Table 1; S1 table ).
The Author F. Tiné collected the data from all the reports.

Data items 10a
List and define all outcomes for which data were sought.Specify whether all results that were compatible with each outcome domain in each study were sought (e.g. for all measures, time points, analyses), and if not, the methods used to decide which results to collect. 10b List and define all other variables for which data were sought (e.g.participant and intervention characteristics, funding sources).Describe any assumptions made about any missing or unclear information.

Study risk of bias assessment
11 Specify the methods used to assess risk of bias in the included studies, including details of the tool(s) used, how many reviewers assessed each study and whether they worked independently, and if applicable, details of automation tools used in the process.
We assessed the covariate imbalance among the trials of each dataset by the method proposed

Section and Topic
Item # Checklist item Location where item is reported in the paper.Rows: 107-160.
Effect measures 12 Specify for each outcome the effect measure(s) (e.g.risk ratio, mean difference) used in the synthesis or presentation of results.
The probabilities of occlusive vascular events and of sustained response.Rows: 321-322.

Synthesis methods 13a
Describe the processes used to decide which studies were eligible for each synthesis (e.g.tabulating the study intervention characteristics and comparing against the planned groups for each synthesis (item #5)).
13b Describe any methods required to prepare the data for presentation or synthesis, such as handling of missing summary statistics, or data conversions.
13c Describe any methods used to tabulate or visually display results of individual studies and syntheses.
13d Describe any methods used to synthesize results and provide a rationale for the choice(s).If meta-analysis was performed, describe the model(s), method(s) to identify the presence and extent of statistical heterogeneity, and software package(s) used.
13e Describe any methods used to explore possible causes of heterogeneity among study results (e.g.subgroup analysis, meta-regression).
13f Describe any sensitivity analyses conducted to assess robustness of the synthesized results.

Reporting bias assessment 14
Describe any methods used to assess risk of bias due to missing results in a synthesis (arising from reporting biases).

Certainty assessment 15
Describe any methods used to assess certainty (or confidence) in the body of evidence for an outcome.

selection 16a
Describe the results of the search and selection process, from the number of records identified in the search to the number of studies included in the review, ideally using a flow diagram.
See the two flow diagrams attached. 16b Cite studies that might appear to meet the inclusion criteria, but which were excluded, and explain why they were excluded.

Study characteristics 17
Cite each included study and present its characteristics.Rows: 163-195

Risk of bias in studies 18
Present assessments of risk of bias for each included study.Rows: 198-307

Results of individual studies 19
For all outcomes, present, for each study: (a) summary statistics for each group (where appropriate) and (b) an effect estimate and its precision (e.g.confidence/credible interval), ideally using structured tables or plots.

Results of syntheses 20a
For each synthesis, briefly summarise the characteristics and risk of bias among contributing studies.
20b Present results of all statistical syntheses conducted.If meta-analysis was done, present for each the summary estimate and its precision (e.g.confidence/credible interval) and measures of statistical heterogeneity.If comparing groups, describe the direction of the effect.
20c Present results of all investigations of possible causes of heterogeneity among study results.
20d Present results of all sensitivity analyses conducted to assess the robustness of the synthesized results.
Reporting biases 21 Present assessments of risk of bias due to missing results (arising from reporting biases) for each synthesis assessed.
Certainty of 22 Present assessments of certainty (or confidence) in the body of evidence for each outcome assessed.
the results for practice, policy, and future research.